Description
operation research applications EDITED BY A. RAVI RAVINDRAN
Submitted to:
Sir Inayatullah
NAMES: SEAT NUMBERS: CLASS:
MISAL UR RUBA QUAZI
AYESHA AFZAL
ANUM AZIZ
FAIZA ISRAFILL
B0916069
B0916023
B0916012
B0916030
BS 4th year
University of Karachi
Topic:
Energy Systems
INDEX
? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ?
INTRODUCTION TO ENERGY Energy Types of energy Future issues of energy Mathematical modeling of energy system Linear Programming Model of Energy Resource Combination (example) Integer Programming Model For Energy Investment Options example Simulation and Optimization of Distributed Energy Systems Point-of-Use Energy Generation Modeling of CHP SYSTEM ECONOMIC OPTIMIZATION METHODS Design of a Model for Optimization of CHP System Capacities Capacity optimization Implementation of a computer model Other Scenarios
INTRODUCTION TO ENERGY
Energy is an all-encompassing commodity that touches the life of everyone ? Kruger (2006) ? Badiru (1982) ? Badiru and Pulat (1995) ? Hudson (2005)
?
ENERGY
A GREEK WORD FOR “WORK.” INDEED, ENERGY IS WHAT MAKES
EVERYTHING WORK. ENERGY
Kinetic Energy
Potential Energy
Kinetic energy is found in anything that moves ? e.g., waves, electrons, atoms, molecules, and physical objects.
?
Potential energy represents stored energy as well as energy of position ? e.g., energy due to fuel, food, and gravity.
?
ENERGY CAN NEIGHER BE CREATED NOR DESTROYED BUT IT TRANSFORMS FROM ONE FORM TO ANOTHER FORM
TYPES OF KINETIC ENERGY
Electrical energy ? Radiant energy ? Motion energy ? Thermal or heat energy ? Sound energy
?
?
TYPES OF POTENTIAL ENERGY
Chemical energy ? Mechanical energy ? Nuclear energy ? Gravitational energy
?
FUTURE ISSUES OF ENERGY
The future of energy will involve several integrative decision scenarios involving technical and managerial issues such as:
Micro power generation systems ? Nega watt systems ? Energy supply transitions ? Coordination of energy alternatives ? Global energy competition ? Green-power generation systems ? Integrative harnessing of sun, wind, and water energy sources ? Energy generation, transformation, transmission, distribution, storage, and consumption across global boundaries.
?
MATHEMATICAL MODELING OF ENERGY SYSTEM
SOME of the mathematical modeling options for energy profiles are: ? linear growth model ? exponential growth model ? logistic curve growth model ? regression model ? logarithmic functions.
LINEAR PROGRAMMING MODEL OF ENERGY RESOURCE COMBINATION
Example:
{This example illustrates the use of LP for energy resource allocation (Badiru, 1982).}
The objective is to find an optimal mix of three different sources of energy to meet the plant’s energy requirements. The three energy sources are • Natural gas Currently being used • Commercial electric power grid • Solar power It is required that the energy mix yield the lowest possible total annual cost of energy for the plant.
• Suppose a forecasting analysis indicates that the minimum kilowatt-hour (kwh) needed per year for heating, cooling, and power are 1,800,000, 1,200,000, and 900,000 kwh , respectively. • The solar energy system is expected to supply at least 1,075,000 kwh annually. The annual use of commercial electric grid must be at least 1,900,000 kwh due to a prevailing contractual agreement for energy supply. The annual consumption of the contracted supply of gas must be at least 950,000 kwh. • The cubic foot unit for natural gas has been converted to kwh (1 cu. ft. of gas=0.3024 kwh). • The respective rates of $6/kwh, $3/kwh, and $2/kwh are applicable to the three sources of energy. • The minimum individual annual conservation credits desired are $600,000 from solar power, $800,000 from commercial electricity, and $375,000 from natural gas. • The conservation credits are associated with the operating and maintenance costs. The energy cost per kilowatt-hour is $0.30 for commercial electricity, $0.05 for natural gas, and $0.40 for solar power. • The initial cost of the solar energy system has been spread over its useful life of 10 years with appropriate cost adjustments to obtain the rate per kilowatt-hour.
ENERGY RESOURCE COMBINATION DATA Energy Source Minimum Supply (1000’s kwh) 1075 Minimum Conservation Credit (1000’$) 600 6 Conservation Credit Rate ($/kwh) Unit Cost ($/kwh) 0.40
Solar Power
Electricity Grid
Natural Gas
1900
950
800
375
3
2
0.30
0.05
Tabulation Of Data For LP Model
Energy Sources
Solar Power Electric Power Natural Gas Constraints
Heating
??11 ??21 ??31 ? 1800
Types Of Use Cooling
??12 ??22 ??32 ? 1200 ?900
Power
??13 ??23 ??33
Constraints
? 1075K ? 1900K ? 950K --------
OBJECTIVE FUNCTION: The optimization problem involves the minimization of the total cost function, Z.
Minimize Z = 0.4 Subject to:
3 ??=1 ??1??
+ 0.3
3 ??=1 ??2??
+ 0.05
3 ??=1 ??3??
??11 +??21 +??31 ? 1800 ??12 +??22 +??32 ? 1200 ??13 +??23 +??33 ? 900 6(??11 +??12 +??13) ? 600 3(??21 +??22 +??23 ) ? 800 2(??31 +??32 +??33 ) ? 375 ??11 +??12 +??13 ? 1075 ??21 +??22 +??23 ? 1900 ??31 +??32 +??33 ? 950 xij ? 0, i, j =1, 2, 3
Solution :
Using the LINDO LP SOFTWARE ,the solution presented below was obtained Energy Sources Heating Cooling Power Solar Power Commercial Electricity Natural Gas 0 975 825 1075 0 125 0 925 0
MIN Z = $1047.50 (in thousands).
INTEGER PROGRAMMING MODEL FOR ENERGY INVESTMENT OPTIONS
•
Integer programming formulation is to maximize present worth of total return on
•
investment. Suppose an energy analyst is given N energy investment options,
??1 , ??2, ??3 , … . . , ???? The investment in each option start at base ???? (i = 1,2,3,…, N) It increases by variable increments ?????? ?? = 1,2, … . , ???? The level of investment in option ???? is
• • •
???? = ???? +
• •
???? ??=?? ??????
where,
???? ? ?? ???
For most cases, the base investment will be 0. In those cases, we will have ???? =0. we have: ?? ???? ?????? ???????????????????? ???? ???????????? ?? ???? ?????????????? ???????? ???????? ?? ?????????????????? ?? ???? ?????? ?????????????????? ???? ?????????????????????? ?? ???? ???????? ?????? = ?? ??????????????????
?????? =
?????
???? ???e actual value of investment in option i, ????? is an indicator variable indicating whether or not option ?? is one of the option is selected for investment. ??? ??????? is the actual magnitude of the ?? increment. ??? ??? increment is used for option i, ???? is an indicator variable that indicates whether or not the ??
The maximum possible investment in each option is defined as, ???? ? ???? ? ???? There is a specified limit B, on the total budget available to invest such that ???? ? ?? For a given energy investment option , the utility function is used to determine the expected return R(???? ) for a specified level of investment in that option. That is: R(???? ) = f(???? ) R(???? ) =
???? ??=1
?????? ??????
Where ?????? is the increment return obtained when the investment in option ?? is increased by ??????.
If the Incremental return decreases as the level of investment increases. The utility function will be concave. We have the following relationship: ?????? ? ????,??+1 ? 0 Thus ?????? ? ????,??+1 ?????? ? ????,??+1 ? 0
9 8
R(x) Curve
h4
7
6
h3 5 4 h2 3 y5
2
1
h1 y2
y3
y4
0 Utility Curve For Investment Yield
This figure shows an example of a concave investment utility function.
Our objective is to maximize the total investment return. That is, Maximize Z = Subject to:
?? ?? ??????
??????
???? = ???? +
?? ??????
??????
???
??? ??? ,j
???? ? ???? ? ???? ?????? ? ????,??+1
?? ????
? ??
???? ?0 ?????? =0 or 1
??? ??? ,j
Example
?
? ? ?
Now suppose we are given four options (i.e., N =4) and a budget limit of $10 million. All the values are in millions of $ Dollar. The respective investments and returns are shown in Tables. to determine how many investment increments should be used for each option
TABLE 5.4
Stage(j)
0 1 2 0.80 0.20
INVESTMENT DATA FOR ENERGY OPTION 1
Level Of Investment ??1
0 0.80 1.00
Incremental Investment ??1??
------
Incremental Return ??1??
-----1.40 0.20
Total Return R(??1 )
0 1.40 1.60
3
4 5
0.20
0.20 0.20
1.20
1.40 1.60
0.30
0.10 0.10
1.90
2.00 2.10
TABLE 5.5
Stage(j) 1 2 3 4 5 6 7
INVESTMENT DATA FOR ENERGY OPTION 2
Incremental Investment ??2?? 3.20 0.20 0.20 0.20 0.20 0.20 0.20 Level Of Investment ??2 3.20 3.40 3.60 3.80 4.00 4.20 4.40 Incremental Return ??2?? 6.00 0.30 0.30 0.20 0.10 0.05 0.05 Total Return R(??2 ) 6.00 6.30 6.60 6.80 6.90 6.90 7.00
TABLE 5.6
Stage(j)
0 1 2 3 4 5 6 7 8 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
INVESTMENT DATA FOR ENERGY OPTION 3
Incremental Investment ??3??
------
Level Of Investment ??3
0 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40
Incremental Return ??3??
-----4.90 0.30 0.40 0.30 0.20 0.10 0.10 0.10
Total Return R(??3 )
0 4.90 5.20 5.60 5.90 6.10 6.20 6.30 6.40
TABLE 5.7
Stage(j) 0 1 2 3 4 5 6 7 1.95 0.20 0.20 0.20 0.20 0.20 0.20
INVESTMENT DATA FOR ENERGY OPTION 4
Incremental Investment ?????? -----Level Of Investment ???? 0 1.95 2.15 2.35 2.55 2.75 2.95 3.15 Incremental Return ?????? -----3.00 0.50 0.20 0.10 0.05 0.15 0.00 Total Return R(???? ) 0 3.00 3.50 3.70 3.80 3.85 4.00 4.00
The IP model of the example was solved with LINDO software.
The model is:
Maximize Z = 1.4 ?????? + .2 ?????? + .3 ?????? + .1 ?????? + .1 ?????? + 6 ?????? + .3 ?????? + .3 ?????? + .2 ?????? + .1 ?????? + .05 ?????? + .05 ?????? + 4.9 ?????? + .3 ?????? + .4 ?????? + .3 ?????? + .2 ?????? + .1 ?????? + .1 ?????? + .1 ?????? + 3 ?????? + .5 ?????? + .2 ?????? + .1 ?????? + .05 ?????? + .15 ??????
Subjected to:
? ? ? ? ?
0.8 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ? ???? = 0
3.2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ????? = 0 2.0 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ? ???? = 0 1.95 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ? ???? = 0 ???? + ???? + ???? + ???? ? 10
?????? ?????? ?????? ??????
? ?????? ? ?????? ? ?????? ? ??????
?0 ?0 ?0 ?0
??????
?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ???? ? 0 for i = 1, 2, . . ., 4 ?????? = 0, 1 for all i and j
The solution indicates the following values for ??????
Energy Option1
Energy Option3
Y???? =1, Y???? =1, Y???? =1, Y???? =1, Y???? =0, Y???? =0, Y???? =0 investment is X3=$2.60 million. return is $5.90 million.
?????? =1, ?????? =1, ?????? =1, ?????? =0, ?????? =0 investment is X1=$1.20 million. return is $1.90 million
Energy Option 2
?????? =1, ?????? =1, ?????? =1, ?????? =1, ?????? =0, ?????? =0, ?????? =0 investment is X2=$3.80 million return is $6.80 million.
Energy Option4
?????? =1, ?????? =1, ?????? =1 investment is X4=$2.35 million. return is $3.70 million.
RESULT:
? ?
?
The total investment in all the four options is $9,950,000. Thus the optimal solution indicates that not all of the $10,000,000 available should be invested . The expected present worth of return from the total investment is $18,300,000. This translates into 83.92% return on investment.
SIMULATION AND OPTIMIZATION OF DISTRIBUTED ENERGY SYSTEMS
? Hudson
(2005) has developed a nonlinear optimization model to determine the optimal capacities of equipment in CHP applications, often called cogeneration. ? If the electrical and thermal load behavior of a facility is given, the cost of the system can be determined. ? CHP application are used to produce optimal cost benefits and potentially avoid economic losses.
POINT-OF-USE ENERGY GENERATION
?
Distributed energy is the provision of energy services at or near the point of use. It can take many forms, but a central element is the existence of a prime mover for generating electricity. When the waste heat is used to satisfy heating needs, the system is typically termed a cogeneration or CHP. CHP systems are merely alternatives to utility grid-supplied electricity, electric chillers, and electric or gas-fired on-site water heating.
?
?
?
?
Through the use of an absorption chiller, waste heat can also be utilized to provide useful cooling.
CHP systems “make-or-buy” decision.
?
MODELING OF CHP SYSTEM
?
?
To optimize a CHP system,simulate the interaction of the system with its loads. The work of different researchers who served to develop different optimization models by simulating the interaction of the system with its loads are explained in the next slide…..
RESEARCHER Baughman et al.
YEAR 1989
ACCOMPLISHMENT developed a cogeneration simulation model in Microsoft Excel using the minimization of net present value of operating costs as the objective function. His model was a mixed integer linear program with an objective function of maximizing hourly profits from operation. introduced a coupled, “hierarchical” modeling concept, optimization of a system’s installed capacity was an outer shell serving to drive a separate inner operations optimization model based on mixed integer L.P. considered the impact of time-of-use rates on optimal sizing and operations of cogeneration systems approach relied upon single unit prices for electricity (i.e., no separate demand charges) and assumed independent Gaussian distributions to describe aggregate thermal and electrical demand. investigated the impact that variation in end-use energy demands had on optimization results. Modeling demand variation as a continuous random variable, probability distributions of electrical and thermal demands were developed. Dividing the problem into discrete elements ,a piecewise LP approach was used to find the minimum cost objective function. developed procedures to evaluate a predefined list of candidate system capacities for an office building application, selecting the CHP system with the minimum net annual cost. selected a reverse approach in investigating capacity-related optimization by varying the building size for a CHP system of fixed capacity. showed that annual energy cost savings exhibit a concave behavior with respect to generator capacity.
Consonni et al. Yokoyama et al.
1989 1991
Asano et al. Wong-Kcomt
1992 1992
Gamou et al.
2002
Marantan
2002
Campanari et al. Czachorski et al.
2002 2002
ECONOMIC OPTIMIZATION METHODS
?
To determine an appropriate optimization method , must have some knowledge of the system .
Initial step is to determine whether the system is linear or nonlinear in either its objective function or its constraints
If Objective function/Constraints are linear Objective function/Constraints are non-linear Indep. variables are constrained Indep. Variables are integers Certain values of indep variables are constrained Max F(x) subjects to ???? ? ?? Then Linear optimization method Non- Linear optimization method Constrained optimization method Specialized integer prog method Convert to unconstrained (via lagrange multipliers etc) Max F(x) subjects to ???? ? ???? ?? = ??
?
• o o o o o o o
Number of methods available for optimization . Some are : Reduced gradient method Newton method
Line search method
Steepest Descent method Conjugate gradient method Line search method Direct search method
o
Sequential search method
REDUCED GRADIENT METHOD : •Converting equality-constrained methods to inequalities. •Reduce number of degree of freedom. CONJUGATE GRADIENT METHOD: •Useful for large problems.
STEEPEST DESCENT METHOD: •Use to solve differentiable functions. •Always the negative gradient.
NEWTON METHOD: (QUASI_NEWTON METHOD) •Very efficient to solve unconstrained nonlinear optimization. •Objective function must have continuous first and second derivative. DIRECT SEARCH METHOD: •Polytope or Nelder-Mead method. •Hook and Jeeves method ( non derivative method). SEQUENTIAL SEARCH METHOD: •Fibonacci and Golden section method.
DESIGN OF A MODEL FOR OPTIMIZATION OF CHP SYSTEM CAPACITIES
This section provides a detailed explanation of the simulation model as well as the approach used to determine an optimum set of equipment capacities for a CHP system.
Define initial capacities vector
Perform hourly operations simulation (see figure A)
Stopping criterion achieved?
NO Perform BFGS update and calculate revised input vector YES
Display optimal solution
Overview flow chart for optimization model
Define input parameters
FIGURE A
Simulate non-CHP scenario
Simulate CHP scenario
For each hour, determine operational status of CHP system
Sum hourly costs CHP application to form annual operation costs for CHP and non-CHP scenario
Combine capital and operating costs from net present value (NVP) savings
Input Variables Used in Operation Simulation Model
Variable Typical Units Variable Typical Units
Facility loads Hourly electrical demand (non-cooling related) Hourly heating demand Hourly cooling demand Electric utility prices Demand charge Energy charge Standby charge Standby charge On-site fuel price (LHV basis) Equipment parameters Boiler efficiency (LHV) Conventional chiller COP Absorption chiller (AC) COP Absorption chiller (AC) capacity Present Without units Without units RT $/kW-month $/kW $/kWh-month $/MMBtu kW Btu/h Btu/h
AC minimum output level AC system parasitic electrical load Distributed generation (DG) capacity, net DG electric efficiency (LHV) at full output DG minimum output level DG power/heat ratio Operating and maintenance (O&M) cost Number of DG units DG capital cost AC capital cost General economic parameters Planning horizon Discount rate Effective income tax rate
Percent kW/RT kW Percent Percent Without units $/kWh Units $/kW installed $/RT installed
Years Percent/Years Percent
?
?
? ? ? ? ? ? ? ? ? ? ? ?
?
? ?
Boiler efficiency Conventional chiller COP Absorption chiller COP Absorption chiller capacity AC minimum output level AC system parasitic electric load Distributed generation (DG) capacity DG electric efficiency (LHV) at full output DG minimum output level DG power/heat ratio O&M cost Number of DG units DG capital cost AC capital cost Planning horizon Discount rate Effective income tax
CAPACITY OPTIMIZATION ? The optimization goal is to maximize the NPV cost savings by determining the optimum installed capacities for the electricity generation system and the absorption chiller. ? Quasi-Newton methods is best to use with BFGS updates of the inverse Hessian matrix without calculating 1st or 2nd order derivatives.
QUASI-NEWTON METHODS
given starting point ???? ? domain f, ???? > 0 ? for k = 1, 2, . . ., until a stopping criterion is satisfied
?
compute quasi-Newton direction ? x = ?????+?? ??? ???? 2. set step size ? =1 (e.g., by backtracking line search) 3. compute ???? = ??????? + ? ? x 4. compute ???? ? If the gradient of the objective function is available ==> ?????? ?????????????? ???????????????????? ??????????? to estimate the gradient ?? vector by ???? = [F(???? +h) - F(???? )]
1.
??
?
different methods use different rules for updating H in step4
?
can also propagate ???? ??? to simplify calculation of ?x
Quasi-Newton Methods (BFGS)
The BFGS update method is considered to be the most efficient and robust approach available in this time The BFGS formula for ????+?? as represented by ZHANG and XU (2001) and BARTHOLOMEW-BIGSS (2005) IS;
????+?? = ???? -
???? ???? ???? ?? +???? ???? ?? ???? ???? ?? ????
+[1 +
???? ?? ???? ???? ???? ???? ?? ???? ?? ????
]
???? ?? ????
38
EXAMPLE OF STEEPEST DECENT METHOD IN THE CASE N= 2.
? ? ? ? ? ? ? ? ?
Let f(x) = ???? H x + ???? b
??
??
with ?? =
?? ?? ; ?? ??
b=
?? ??
Let ???? = [2 1]?? and ???? = ????×?? ? x = ????? ??? ???? = [?8 ?6]?? A line search to minimize f(???? + ? ? x) = 180??? ?100 ?+18 yields minimum at ? =
?? ???? ?? ???? ?? ??? ?? ?? ??
The next iterate is ???? = ???? +
? x = [?
]
Up to this point it’s just steepest descent. Now we need to find an updated Hessian ???? . ?? ?????? It must be symmetric, so that H = ???? ?????? ?????? The Quasi-Newton condition also comes into play. We have
????? ?????
?
? ???? = ???? ? ???? =
?? ??? ??
; ? ???? = ???? ? ???? =
?? ????? ??
?
The Quasi-Newton condition (note k= 0 still, so we only need to consider the case i= 0 in equation (2)) yields equations
????? ??
? ? ?
?????? ?
????? ??
?????? =
????? ??
;
????? ??
?????? ?
????? ??
?????? =
??? ??
This is two equations in three unknowns, so we ought to have a free variable. Take ?????? arbitrarily With ?????? = 1 we find ?????? =?
???? ????
?
and ?????? =
???? ????
?? H= ?
???? ????
?
????
???? ???? ????
and the iteration continues….
IMPLEMENTATION OF THE COMPUTER MODEL
To provide useful transparency of the calculations, the methods were implemented using Microsoft Excel. ? Example: • A hospital in Boston, consists of five stories with a total floor area of 500,000 square feet. • The maximum electrical load is 2275 kW; the maximum heating load is 17 million Btu/h; and the maximum cooling load is 808 refrigeration tons (RT). • Hourly electrical and thermal demands for the facility were obtained using the building simulator program. • The price of natural gas in the initial year of operation was assumed to be $11.00/million Btu. • Escalation assumptions, expressed in percent change from the previous year, for this example are provided in Table 5.9.
•
FIGURE 5.6
Optimization results for a Boston hospital.
OTHER SCENARIOS
Due to the variation in fuel and electricity prices, site weather, and electrical and thermal loads of a specific facility, each potential CHP application will have a unique optimal solution that maximizes economic benefit. ? To maximize the economic benefit of a CHP system, each application should be individually evaluated by the methods described in this chapter.
?
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doc_650728478.pptx
operation research applications EDITED BY A. RAVI RAVINDRAN
Submitted to:
Sir Inayatullah
NAMES: SEAT NUMBERS: CLASS:
MISAL UR RUBA QUAZI
AYESHA AFZAL
ANUM AZIZ
FAIZA ISRAFILL
B0916069
B0916023
B0916012
B0916030
BS 4th year
University of Karachi
Topic:
Energy Systems
INDEX
? ? ? ? ?
?
? ? ? ? ? ? ? ? ? ?
INTRODUCTION TO ENERGY Energy Types of energy Future issues of energy Mathematical modeling of energy system Linear Programming Model of Energy Resource Combination (example) Integer Programming Model For Energy Investment Options example Simulation and Optimization of Distributed Energy Systems Point-of-Use Energy Generation Modeling of CHP SYSTEM ECONOMIC OPTIMIZATION METHODS Design of a Model for Optimization of CHP System Capacities Capacity optimization Implementation of a computer model Other Scenarios
INTRODUCTION TO ENERGY
Energy is an all-encompassing commodity that touches the life of everyone ? Kruger (2006) ? Badiru (1982) ? Badiru and Pulat (1995) ? Hudson (2005)
?
ENERGY
A GREEK WORD FOR “WORK.” INDEED, ENERGY IS WHAT MAKES
EVERYTHING WORK. ENERGY
Kinetic Energy
Potential Energy
Kinetic energy is found in anything that moves ? e.g., waves, electrons, atoms, molecules, and physical objects.
?
Potential energy represents stored energy as well as energy of position ? e.g., energy due to fuel, food, and gravity.
?
ENERGY CAN NEIGHER BE CREATED NOR DESTROYED BUT IT TRANSFORMS FROM ONE FORM TO ANOTHER FORM
TYPES OF KINETIC ENERGY
Electrical energy ? Radiant energy ? Motion energy ? Thermal or heat energy ? Sound energy
?
?
TYPES OF POTENTIAL ENERGY
Chemical energy ? Mechanical energy ? Nuclear energy ? Gravitational energy
?
FUTURE ISSUES OF ENERGY
The future of energy will involve several integrative decision scenarios involving technical and managerial issues such as:
Micro power generation systems ? Nega watt systems ? Energy supply transitions ? Coordination of energy alternatives ? Global energy competition ? Green-power generation systems ? Integrative harnessing of sun, wind, and water energy sources ? Energy generation, transformation, transmission, distribution, storage, and consumption across global boundaries.
?
MATHEMATICAL MODELING OF ENERGY SYSTEM
SOME of the mathematical modeling options for energy profiles are: ? linear growth model ? exponential growth model ? logistic curve growth model ? regression model ? logarithmic functions.
LINEAR PROGRAMMING MODEL OF ENERGY RESOURCE COMBINATION
Example:
{This example illustrates the use of LP for energy resource allocation (Badiru, 1982).}
The objective is to find an optimal mix of three different sources of energy to meet the plant’s energy requirements. The three energy sources are • Natural gas Currently being used • Commercial electric power grid • Solar power It is required that the energy mix yield the lowest possible total annual cost of energy for the plant.
• Suppose a forecasting analysis indicates that the minimum kilowatt-hour (kwh) needed per year for heating, cooling, and power are 1,800,000, 1,200,000, and 900,000 kwh , respectively. • The solar energy system is expected to supply at least 1,075,000 kwh annually. The annual use of commercial electric grid must be at least 1,900,000 kwh due to a prevailing contractual agreement for energy supply. The annual consumption of the contracted supply of gas must be at least 950,000 kwh. • The cubic foot unit for natural gas has been converted to kwh (1 cu. ft. of gas=0.3024 kwh). • The respective rates of $6/kwh, $3/kwh, and $2/kwh are applicable to the three sources of energy. • The minimum individual annual conservation credits desired are $600,000 from solar power, $800,000 from commercial electricity, and $375,000 from natural gas. • The conservation credits are associated with the operating and maintenance costs. The energy cost per kilowatt-hour is $0.30 for commercial electricity, $0.05 for natural gas, and $0.40 for solar power. • The initial cost of the solar energy system has been spread over its useful life of 10 years with appropriate cost adjustments to obtain the rate per kilowatt-hour.
ENERGY RESOURCE COMBINATION DATA Energy Source Minimum Supply (1000’s kwh) 1075 Minimum Conservation Credit (1000’$) 600 6 Conservation Credit Rate ($/kwh) Unit Cost ($/kwh) 0.40
Solar Power
Electricity Grid
Natural Gas
1900
950
800
375
3
2
0.30
0.05
Tabulation Of Data For LP Model
Energy Sources
Solar Power Electric Power Natural Gas Constraints
Heating
??11 ??21 ??31 ? 1800
Types Of Use Cooling
??12 ??22 ??32 ? 1200 ?900
Power
??13 ??23 ??33
Constraints
? 1075K ? 1900K ? 950K --------
OBJECTIVE FUNCTION: The optimization problem involves the minimization of the total cost function, Z.
Minimize Z = 0.4 Subject to:
3 ??=1 ??1??
+ 0.3
3 ??=1 ??2??
+ 0.05
3 ??=1 ??3??
??11 +??21 +??31 ? 1800 ??12 +??22 +??32 ? 1200 ??13 +??23 +??33 ? 900 6(??11 +??12 +??13) ? 600 3(??21 +??22 +??23 ) ? 800 2(??31 +??32 +??33 ) ? 375 ??11 +??12 +??13 ? 1075 ??21 +??22 +??23 ? 1900 ??31 +??32 +??33 ? 950 xij ? 0, i, j =1, 2, 3
Solution :
Using the LINDO LP SOFTWARE ,the solution presented below was obtained Energy Sources Heating Cooling Power Solar Power Commercial Electricity Natural Gas 0 975 825 1075 0 125 0 925 0
MIN Z = $1047.50 (in thousands).
INTEGER PROGRAMMING MODEL FOR ENERGY INVESTMENT OPTIONS
•
Integer programming formulation is to maximize present worth of total return on
•
investment. Suppose an energy analyst is given N energy investment options,
??1 , ??2, ??3 , … . . , ???? The investment in each option start at base ???? (i = 1,2,3,…, N) It increases by variable increments ?????? ?? = 1,2, … . , ???? The level of investment in option ???? is
• • •
???? = ???? +
• •
???? ??=?? ??????
where,
???? ? ?? ???
For most cases, the base investment will be 0. In those cases, we will have ???? =0. we have: ?? ???? ?????? ???????????????????? ???? ???????????? ?? ???? ?????????????? ???????? ???????? ?? ?????????????????? ?? ???? ?????? ?????????????????? ???? ?????????????????????? ?? ???? ???????? ?????? = ?? ??????????????????
?????? =
?????
???? ???e actual value of investment in option i, ????? is an indicator variable indicating whether or not option ?? is one of the option is selected for investment. ??? ??????? is the actual magnitude of the ?? increment. ??? ??? increment is used for option i, ???? is an indicator variable that indicates whether or not the ??
The maximum possible investment in each option is defined as, ???? ? ???? ? ???? There is a specified limit B, on the total budget available to invest such that ???? ? ?? For a given energy investment option , the utility function is used to determine the expected return R(???? ) for a specified level of investment in that option. That is: R(???? ) = f(???? ) R(???? ) =
???? ??=1
?????? ??????
Where ?????? is the increment return obtained when the investment in option ?? is increased by ??????.
If the Incremental return decreases as the level of investment increases. The utility function will be concave. We have the following relationship: ?????? ? ????,??+1 ? 0 Thus ?????? ? ????,??+1 ?????? ? ????,??+1 ? 0
9 8
R(x) Curve
h4
7
6
h3 5 4 h2 3 y5
2
1
h1 y2
y3
y4
0 Utility Curve For Investment Yield
This figure shows an example of a concave investment utility function.
Our objective is to maximize the total investment return. That is, Maximize Z = Subject to:
?? ?? ??????
??????
???? = ???? +
?? ??????
??????
???
??? ??? ,j
???? ? ???? ? ???? ?????? ? ????,??+1
?? ????
? ??
???? ?0 ?????? =0 or 1
??? ??? ,j
Example
?
? ? ?
Now suppose we are given four options (i.e., N =4) and a budget limit of $10 million. All the values are in millions of $ Dollar. The respective investments and returns are shown in Tables. to determine how many investment increments should be used for each option
TABLE 5.4
Stage(j)
0 1 2 0.80 0.20
INVESTMENT DATA FOR ENERGY OPTION 1
Level Of Investment ??1
0 0.80 1.00
Incremental Investment ??1??
------
Incremental Return ??1??
-----1.40 0.20
Total Return R(??1 )
0 1.40 1.60
3
4 5
0.20
0.20 0.20
1.20
1.40 1.60
0.30
0.10 0.10
1.90
2.00 2.10
TABLE 5.5
Stage(j) 1 2 3 4 5 6 7
INVESTMENT DATA FOR ENERGY OPTION 2
Incremental Investment ??2?? 3.20 0.20 0.20 0.20 0.20 0.20 0.20 Level Of Investment ??2 3.20 3.40 3.60 3.80 4.00 4.20 4.40 Incremental Return ??2?? 6.00 0.30 0.30 0.20 0.10 0.05 0.05 Total Return R(??2 ) 6.00 6.30 6.60 6.80 6.90 6.90 7.00
TABLE 5.6
Stage(j)
0 1 2 3 4 5 6 7 8 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
INVESTMENT DATA FOR ENERGY OPTION 3
Incremental Investment ??3??
------
Level Of Investment ??3
0 2.00 2.20 2.40 2.60 2.80 3.00 3.20 3.40
Incremental Return ??3??
-----4.90 0.30 0.40 0.30 0.20 0.10 0.10 0.10
Total Return R(??3 )
0 4.90 5.20 5.60 5.90 6.10 6.20 6.30 6.40
TABLE 5.7
Stage(j) 0 1 2 3 4 5 6 7 1.95 0.20 0.20 0.20 0.20 0.20 0.20
INVESTMENT DATA FOR ENERGY OPTION 4
Incremental Investment ?????? -----Level Of Investment ???? 0 1.95 2.15 2.35 2.55 2.75 2.95 3.15 Incremental Return ?????? -----3.00 0.50 0.20 0.10 0.05 0.15 0.00 Total Return R(???? ) 0 3.00 3.50 3.70 3.80 3.85 4.00 4.00
The IP model of the example was solved with LINDO software.
The model is:
Maximize Z = 1.4 ?????? + .2 ?????? + .3 ?????? + .1 ?????? + .1 ?????? + 6 ?????? + .3 ?????? + .3 ?????? + .2 ?????? + .1 ?????? + .05 ?????? + .05 ?????? + 4.9 ?????? + .3 ?????? + .4 ?????? + .3 ?????? + .2 ?????? + .1 ?????? + .1 ?????? + .1 ?????? + 3 ?????? + .5 ?????? + .2 ?????? + .1 ?????? + .05 ?????? + .15 ??????
Subjected to:
? ? ? ? ?
0.8 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ? ???? = 0
3.2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ????? = 0 2.0 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ? ???? = 0 1.95 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? + .2 ?????? ? ???? = 0 ???? + ???? + ???? + ???? ? 10
?????? ?????? ?????? ??????
? ?????? ? ?????? ? ?????? ? ??????
?0 ?0 ?0 ?0
??????
?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ?????? ? ?????? ? 0 ???? ? 0 for i = 1, 2, . . ., 4 ?????? = 0, 1 for all i and j
The solution indicates the following values for ??????
Energy Option1
Energy Option3
Y???? =1, Y???? =1, Y???? =1, Y???? =1, Y???? =0, Y???? =0, Y???? =0 investment is X3=$2.60 million. return is $5.90 million.
?????? =1, ?????? =1, ?????? =1, ?????? =0, ?????? =0 investment is X1=$1.20 million. return is $1.90 million
Energy Option 2
?????? =1, ?????? =1, ?????? =1, ?????? =1, ?????? =0, ?????? =0, ?????? =0 investment is X2=$3.80 million return is $6.80 million.
Energy Option4
?????? =1, ?????? =1, ?????? =1 investment is X4=$2.35 million. return is $3.70 million.
RESULT:
? ?
?
The total investment in all the four options is $9,950,000. Thus the optimal solution indicates that not all of the $10,000,000 available should be invested . The expected present worth of return from the total investment is $18,300,000. This translates into 83.92% return on investment.
SIMULATION AND OPTIMIZATION OF DISTRIBUTED ENERGY SYSTEMS
? Hudson
(2005) has developed a nonlinear optimization model to determine the optimal capacities of equipment in CHP applications, often called cogeneration. ? If the electrical and thermal load behavior of a facility is given, the cost of the system can be determined. ? CHP application are used to produce optimal cost benefits and potentially avoid economic losses.
POINT-OF-USE ENERGY GENERATION
?
Distributed energy is the provision of energy services at or near the point of use. It can take many forms, but a central element is the existence of a prime mover for generating electricity. When the waste heat is used to satisfy heating needs, the system is typically termed a cogeneration or CHP. CHP systems are merely alternatives to utility grid-supplied electricity, electric chillers, and electric or gas-fired on-site water heating.
?
?
?
?
Through the use of an absorption chiller, waste heat can also be utilized to provide useful cooling.
CHP systems “make-or-buy” decision.
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MODELING OF CHP SYSTEM
?
?
To optimize a CHP system,simulate the interaction of the system with its loads. The work of different researchers who served to develop different optimization models by simulating the interaction of the system with its loads are explained in the next slide…..
RESEARCHER Baughman et al.
YEAR 1989
ACCOMPLISHMENT developed a cogeneration simulation model in Microsoft Excel using the minimization of net present value of operating costs as the objective function. His model was a mixed integer linear program with an objective function of maximizing hourly profits from operation. introduced a coupled, “hierarchical” modeling concept, optimization of a system’s installed capacity was an outer shell serving to drive a separate inner operations optimization model based on mixed integer L.P. considered the impact of time-of-use rates on optimal sizing and operations of cogeneration systems approach relied upon single unit prices for electricity (i.e., no separate demand charges) and assumed independent Gaussian distributions to describe aggregate thermal and electrical demand. investigated the impact that variation in end-use energy demands had on optimization results. Modeling demand variation as a continuous random variable, probability distributions of electrical and thermal demands were developed. Dividing the problem into discrete elements ,a piecewise LP approach was used to find the minimum cost objective function. developed procedures to evaluate a predefined list of candidate system capacities for an office building application, selecting the CHP system with the minimum net annual cost. selected a reverse approach in investigating capacity-related optimization by varying the building size for a CHP system of fixed capacity. showed that annual energy cost savings exhibit a concave behavior with respect to generator capacity.
Consonni et al. Yokoyama et al.
1989 1991
Asano et al. Wong-Kcomt
1992 1992
Gamou et al.
2002
Marantan
2002
Campanari et al. Czachorski et al.
2002 2002
ECONOMIC OPTIMIZATION METHODS
?
To determine an appropriate optimization method , must have some knowledge of the system .
Initial step is to determine whether the system is linear or nonlinear in either its objective function or its constraints
If Objective function/Constraints are linear Objective function/Constraints are non-linear Indep. variables are constrained Indep. Variables are integers Certain values of indep variables are constrained Max F(x) subjects to ???? ? ?? Then Linear optimization method Non- Linear optimization method Constrained optimization method Specialized integer prog method Convert to unconstrained (via lagrange multipliers etc) Max F(x) subjects to ???? ? ???? ?? = ??
?
• o o o o o o o
Number of methods available for optimization . Some are : Reduced gradient method Newton method
Line search method
Steepest Descent method Conjugate gradient method Line search method Direct search method
o
Sequential search method
REDUCED GRADIENT METHOD : •Converting equality-constrained methods to inequalities. •Reduce number of degree of freedom. CONJUGATE GRADIENT METHOD: •Useful for large problems.
STEEPEST DESCENT METHOD: •Use to solve differentiable functions. •Always the negative gradient.
NEWTON METHOD: (QUASI_NEWTON METHOD) •Very efficient to solve unconstrained nonlinear optimization. •Objective function must have continuous first and second derivative. DIRECT SEARCH METHOD: •Polytope or Nelder-Mead method. •Hook and Jeeves method ( non derivative method). SEQUENTIAL SEARCH METHOD: •Fibonacci and Golden section method.
DESIGN OF A MODEL FOR OPTIMIZATION OF CHP SYSTEM CAPACITIES
This section provides a detailed explanation of the simulation model as well as the approach used to determine an optimum set of equipment capacities for a CHP system.
Define initial capacities vector
Perform hourly operations simulation (see figure A)
Stopping criterion achieved?
NO Perform BFGS update and calculate revised input vector YES
Display optimal solution
Overview flow chart for optimization model
Define input parameters
FIGURE A
Simulate non-CHP scenario
Simulate CHP scenario
For each hour, determine operational status of CHP system
Sum hourly costs CHP application to form annual operation costs for CHP and non-CHP scenario
Combine capital and operating costs from net present value (NVP) savings
Input Variables Used in Operation Simulation Model
Variable Typical Units Variable Typical Units
Facility loads Hourly electrical demand (non-cooling related) Hourly heating demand Hourly cooling demand Electric utility prices Demand charge Energy charge Standby charge Standby charge On-site fuel price (LHV basis) Equipment parameters Boiler efficiency (LHV) Conventional chiller COP Absorption chiller (AC) COP Absorption chiller (AC) capacity Present Without units Without units RT $/kW-month $/kW $/kWh-month $/MMBtu kW Btu/h Btu/h
AC minimum output level AC system parasitic electrical load Distributed generation (DG) capacity, net DG electric efficiency (LHV) at full output DG minimum output level DG power/heat ratio Operating and maintenance (O&M) cost Number of DG units DG capital cost AC capital cost General economic parameters Planning horizon Discount rate Effective income tax rate
Percent kW/RT kW Percent Percent Without units $/kWh Units $/kW installed $/RT installed
Years Percent/Years Percent
?
?
? ? ? ? ? ? ? ? ? ? ? ?
?
? ?
Boiler efficiency Conventional chiller COP Absorption chiller COP Absorption chiller capacity AC minimum output level AC system parasitic electric load Distributed generation (DG) capacity DG electric efficiency (LHV) at full output DG minimum output level DG power/heat ratio O&M cost Number of DG units DG capital cost AC capital cost Planning horizon Discount rate Effective income tax
CAPACITY OPTIMIZATION ? The optimization goal is to maximize the NPV cost savings by determining the optimum installed capacities for the electricity generation system and the absorption chiller. ? Quasi-Newton methods is best to use with BFGS updates of the inverse Hessian matrix without calculating 1st or 2nd order derivatives.
QUASI-NEWTON METHODS
given starting point ???? ? domain f, ???? > 0 ? for k = 1, 2, . . ., until a stopping criterion is satisfied
?
compute quasi-Newton direction ? x = ?????+?? ??? ???? 2. set step size ? =1 (e.g., by backtracking line search) 3. compute ???? = ??????? + ? ? x 4. compute ???? ? If the gradient of the objective function is available ==> ?????? ?????????????? ???????????????????? ??????????? to estimate the gradient ?? vector by ???? = [F(???? +h) - F(???? )]
1.
??
?
different methods use different rules for updating H in step4
?
can also propagate ???? ??? to simplify calculation of ?x
Quasi-Newton Methods (BFGS)
The BFGS update method is considered to be the most efficient and robust approach available in this time The BFGS formula for ????+?? as represented by ZHANG and XU (2001) and BARTHOLOMEW-BIGSS (2005) IS;
????+?? = ???? -
???? ???? ???? ?? +???? ???? ?? ???? ???? ?? ????
+[1 +
???? ?? ???? ???? ???? ???? ?? ???? ?? ????
]
???? ?? ????
38
EXAMPLE OF STEEPEST DECENT METHOD IN THE CASE N= 2.
? ? ? ? ? ? ? ? ?
Let f(x) = ???? H x + ???? b
??
??
with ?? =
?? ?? ; ?? ??
b=
?? ??
Let ???? = [2 1]?? and ???? = ????×?? ? x = ????? ??? ???? = [?8 ?6]?? A line search to minimize f(???? + ? ? x) = 180??? ?100 ?+18 yields minimum at ? =
?? ???? ?? ???? ?? ??? ?? ?? ??
The next iterate is ???? = ???? +
? x = [?
]
Up to this point it’s just steepest descent. Now we need to find an updated Hessian ???? . ?? ?????? It must be symmetric, so that H = ???? ?????? ?????? The Quasi-Newton condition also comes into play. We have
????? ?????
?
? ???? = ???? ? ???? =
?? ??? ??
; ? ???? = ???? ? ???? =
?? ????? ??
?
The Quasi-Newton condition (note k= 0 still, so we only need to consider the case i= 0 in equation (2)) yields equations
????? ??
? ? ?
?????? ?
????? ??
?????? =
????? ??
;
????? ??
?????? ?
????? ??
?????? =
??? ??
This is two equations in three unknowns, so we ought to have a free variable. Take ?????? arbitrarily With ?????? = 1 we find ?????? =?
???? ????
?
and ?????? =
???? ????
?? H= ?
???? ????
?
????
???? ???? ????
and the iteration continues….
IMPLEMENTATION OF THE COMPUTER MODEL
To provide useful transparency of the calculations, the methods were implemented using Microsoft Excel. ? Example: • A hospital in Boston, consists of five stories with a total floor area of 500,000 square feet. • The maximum electrical load is 2275 kW; the maximum heating load is 17 million Btu/h; and the maximum cooling load is 808 refrigeration tons (RT). • Hourly electrical and thermal demands for the facility were obtained using the building simulator program. • The price of natural gas in the initial year of operation was assumed to be $11.00/million Btu. • Escalation assumptions, expressed in percent change from the previous year, for this example are provided in Table 5.9.
•
FIGURE 5.6
Optimization results for a Boston hospital.
OTHER SCENARIOS
Due to the variation in fuel and electricity prices, site weather, and electrical and thermal loads of a specific facility, each potential CHP application will have a unique optimal solution that maximizes economic benefit. ? To maximize the economic benefit of a CHP system, each application should be individually evaluated by the methods described in this chapter.
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